SOLUTION OF THE FOKKER- PLANCK EQUATION

CHAPTER II CHAPTER II

2.1 SOLUTION OF THE FOKKER- PLANCK EQUATION

Piecewise linear systems

In this chapter a method is derived to obtain the spectral density and other properties of the system (0 .1), where the functions f(x), h.(x) are piecewise linear. By this is meant that they are linear

in each of a finite number (say n-1) of segments (xi+l' xi); the endpoints x

1, xn of the interval in which the process occurs may or may not be finite, and f and h. need not be continuous at the points Xi• In

the ith segment, (xi+l' xi)' we put f(x)

= {. .

x + k .

01 01 (2. 1)

h. (x) = -r.. .. x

+

k ..

J J1 J1 (2. 2)

The FP equation

For the first order system (0.1 ), the FP equation (1.11) for P(x, t Ix ) becomes

ap a

² ⁽ ^· ^'

a ( )

&f" = - -

₂ a(x)P) -

ox

^b{x)P ^,

ox

^{(2. 3)}

where, substituting (2.1), (2.2) into (1.15), (1.17),

a(x)

=

^L: D.k(,f, .. x+k .. )(,f,k.x+kk.)

. k J Jl Jl 1 1

b(x) =-,{', .x-k . + L: D.k,f, .. (..L .x+kk.)

01 01 . k J Jl -kl 1

for x E (xi+l' xi). Thus a(x}, b(x) can be written a(x} = a.(x) = A.x + 2B.x + C. 2

1 l l 1

b(x}

=

bi (x)

=

^D.x+ E.

1 1

for x E (xi+l' xi)' where Ai, Bi, Ci, Di' Ei (i

=

1, .•. , n-1) are constants. Since the matrix Djk is non-negative, (2. 4) gives

A.G.

~

B.2

for every i

1 1 1

(2. 4) (2. 5)

(2. 6) (2. 7)

(2. 8)

It is convenient to include all irregular points--i. e. where a(x) __, 0 on one or both sides - -among the endpoints x. of intervals.

. 1

The backwards equation

This is the formal adjoint of (2. 3 ), namely

ap

=

Initial conditions

a(x ) - -o2P

2 + b(x )

ox

⁰

ax . op

(2. 9)

The initial condition for both (2.3) and (2.9) is, from (1.12), P(x, 0 jx )

=

6 (x-x )

0 0 (2. 1 0)

Boundary conditions

At x1 and xn' a reflecting boundary is assumed (see

section 1. 4). Thus the appropriate boundary conditions for (2. 3) are

Q(x1, t Ix ) _o

=

^{Q(x ,}_n ^{t Jx}_o⁾

=

^{0 ,} ^(2.^{11 )}

where

= ox o

(aP) - bP , (2. 1 2)

and the appropriate boundary conditions for (2. 9) are

oP op I

, a;;-

(x, t jx

= a;z-

(x, t xn) = 0 .

0 0

(2. 13)

Conditions (2. 11) and (2. 13) are both necessary only if x

1 and xn are regular boundaries. If x

1 or xn is infinite, it is a natural boundary, and (2. 11) and (2. 13) are both automatically satisfied. If x1 or xn is finite but irregular, one or the other condition, or both, will be redundant (according to whether it is an entrance, exit or natural boundary). Note that one is restricted to integrable solutions of (2. 3) and bounded solutions of (2. 9).

Since the method of solution is to find the general solution in each interval of linearity, and to piece these together, boundary conditions are also needed at all points x. (i=2, ••• ,n-1). These are

provided for (2. 3) by the continuity of Q and aP with respect to x, and for (2. 9) by the continuity of P and-:.!- with respect to x • Again

x0 O

some conditions become redundant when the x. are irregular points.

The Laplace-transformed FP equation

By applying a Laplace transform with respect to t to (2. 3 ), one obtains a second order ordinary differential equation for

p(x, s jx ), the Laplace transform of P(x, t Ix ) • This is (using the

0 0

initial condition)

: 2 (a(x)p )-

;!

(b(x)p) - sp

=

^- o(x-x ) .

0 (2.14)

Similarly the Laplace transform of the backwards equation (2. 9) is (2.15)

The boundary conditions remain the same as in the untrans- formed case, except that P and Q are replaced by their transforms, namely p and q.

Singularities of the transformed FF operator

Since (for x E (xi+l' xi)) a(x) is a quadratic in x while b(x) is linear, the transformed FF operator

2 .

---:;:

_:r·-

= a(x) dxd [ I

J

^d ^[ ^II ^I

^l

+

2a (x)- b(x) dx

+

a (x)-b (x)-s J ( .) (2. 16) will have 3 regular singular points in the complex plane, at co and at

- B.±[B.2

-A.C.] ²

1 1 1 1

A. l

(2. 1 7)

except when 2 or 3 of these points coalesce to form an irregular singularity. Thus 2 linearly independent solutions for the homogeneous equation a<':'p

=

0 can be found in terms of known special functions.

The following cases occur.

(a) A.

1

B. 2 < A.C . .

l 1 l Then

a.

1 and

a.

2 are distinct and finite. By a linear transformation of the independent variable which transforms

a.

1 to 0 and

°'z

to l, the equation becomes the hypergeometric equation. 1 (b) . A.

1

B. 2

=

A.C .•

1 1 1 Then

a.

1 and

a.

2 coalesce to form

( c)

an irregular (double) singularity. By replacing the independent variable x by (x+B.)-1

, the equation

becomes a confluent hypergeometric equation. 2 A . = B. = O.

l 1 Then

a.

1 and o.

2 both coalesce with ro, forming an irregular (triple) singularity. The solution can be expressed in terms of parabolic cylinder

f . 3

unctions.

The case A.

=

0, B.

1

0 cannot occur, on account of (2. 8). This

l l .

also shows that

o.

1 and

a.

2 are not real unless they are equal, so that the standard solutions for case (a) are complex-valued and must be combined to form 2 independent real solutions (since only real solutions are of interest in what follows). Except in special cases, this leads to considerable algebraic complication. One such special case is dealt with in section 5. 4. Case (b) is dealt with in Chapter IV 1 For solutions of this equation, see Abramowitz and Stegun

[1], page 563.

2 For solutions see Erdelyi et al. [11], page 251.

3 See [1], page 686.

(a special case) and sections 5. 2-3 (a less special case), while

case (c) (forcing function excitation only) is dealt with in Chapter III.

Although the above deals only with the FP operator, it is also true of the transformed backwards operator 3'. This remark

applies also to the rest of this section, mutatis mutandis.

General solution in any sub-interval

As indicated above, it is possible to find, in terms of known special functions, two linearly independent functions p/(x), p

2i(x) which satisfy the homogeneous equation 3'>:<p

=

⁰for x E (x.

1, x.). ·It

i+

will be assumed that p₁\ x) and p;(x) are real for real x. Since the coefficients of the differential equation are real, two such real solutions do exist; if the standard solutions are not real, two linear combinations of them will be.

Then if x

0 ~ (xi, xi+l) the general solution for x in this interval will be

1 1

where c

1 and c

2 are constant with respect to x. If x

0 E (xk, xk+l ), then for x in this interval a particular solution is

I

k k k k

P1 (x)p2 (z) - Pz (x)pl (z)

o(z-x )dz ak(z) wk(z) 0

0 for x

= k k k k

P1 (x)p2 (xo) - P2 (x)pl (xo)

for x ak(xo)wk(xo)

~x 0

(2. 19)

~x 0

Here

k k

k dp2 (x) . k dpl (x)

wk(x)

=

P1 (x) dx - P2 (x) dx (2. 20)

k k

is the Wronskian of the solutions p 1 , p

2 • Then (2. 18) applies also in the interval (xk+l' xk), except that the constants c

1k, c

2k have

k+ k+ k- k-

different values for x > x

0 and x < x

0, say c

1 c

2 and c₁ , c 2 respectively, for which

(2. 21 ) a1 <: ^(x0 ^{)wk(x )}0

(2.22)

When x

=

xi, no difficulty is experienced; be cause of the continuity of p(x, s lx

0) with respect to x

0, this can be considered as the limit of either of the two cases just dealt with.

an irregular point can be dealt with as they arise.

Boundary conditions

Cases when x is

On account of (2. 11), one has at the end points x

1 ^{and xn} (2.23)

0 . ^{(2. 24)}

(Here q.i

=

dxd (a.p.i) - b.p.i.) At the junction points of intervals, J ¹ J ¹ J

x. (i

=

2, .•. , n-1 ), the continuity of Q and aP (and hence q and ap)

shows that

(2. 2 5)

Existence of a solution to the transformed FP equation

Equations (2.21-26) form a set of 2n linear algebraic equations for the 2n otherwise undetermined coefficients

c/,

^c₂¹^,

This nonhomogeneous set will have a unique nontrivial solution provided the

corresponding homogeneous set has no nontrivial solution (the

Fredholm alternative). The only nonhomogeneous terms are those on the right of (2.21) and (2.22). Putting these equal to zero is equivalent to omi tt:lng the term

o

(x-x ) on the right of (2. 1 4), i. e. , to solving the

o .

homogeneous differential equation (with the same boundary conditions).

This will have a nontrivial solution only if s

=

A., where A. is an eigen-

, ,, a

a

value of the original FP operator, ;Ji'''= a}-(a.) -

Bx

(b .). But this can only occur for values of s for which p{x, s

Ix )

does not exist. Thus

the nonhomogeneous set will have a unique nontrivial solution when- ever p(x, s Ix ) exists; which is of course to be expected. In

particular it will have a solution for all Re s > 0.

The transformed transition density Having solved (2. 21-26 ), p{x, s lx

0 ), the Laplace transform of P{x, t \x ), is immediately found using (2 .18). Although the inverse

transform has been found in only a few special cases (when it can usually be found more readily without transforming in the first place),

p(x, s Ix ) is of considerable use as it is.

For example, it can be verified that

lim sp(x, s Ix )

=

P (x}

=

lim P(x, t jx ) ,

s-+O ⁰ ⁰ t-+oo ⁰

(2. 2 7)

where P (x) is found as in section 1. 6. See the following section and

the special cases of Chapters III - V.

As shown in section 2. 3, the Laplace transform of the auto- correlation can be explicitly formulated in terms of p and its deriva- tives with respect to x and x • This can then be used to obtain the

spectral density. This is the principal result in this thesis.

2. 2 THE STEADY -STA TE DENSITY Determination of P

The method of determination of P (x} is described in section

0 .

1. 6. If (xn, x

1) is a regular interval with reflecting or nonattracting natural boundaries, then, from (l .65),

c .

⁰¹

[I (

^D.x+E¹ ¹^. ⁾

^~

P 0 (x)

=

2 exp

2 ^dx

A.x +2B.x+C. A.x +2B.x+C.

1 1 1 1 1 1

(2.28)

The integral here can be evaluated by elementary methods. The continuity of aP allows the determination of the constants C . up to a

0 01

multiplicative factor, which can be found using (1.63), i.e.

x n

(2.29)

For the evaluation of this integral, see below.

If (x

1, xn) contains irregular points, and/or has absorbing or attracting natural boundaries, then the situation becomes more

involved. The steady-state density may depend on x , contain delta

functions, or be identically zero. For its evaluation, see section 1. 6.

The integral of P

In almost all cases, the determination of P (x) requires the

calculation of the indefinite integral

[

~

2 -1 D.z+E.

I(x)

=

J(Aix +2Bix+cJ exp

J (

₂¹ ¹ ⁾ ^dz ^dx

A.z +2B.z+C. · 1

l l l ._J

(2. 3 0)

In some cases, this integral can be found by elementary methods. In general, one notes that J

*

P = 0 can be written

d [ dIJ dl

dx a(x) dx - b(x) dx

=

0 , (2. 31 ) which is a second order equation of the same type as the homogeneous Laplace-transformed FP equation solved in section 2.1. One solution to (2. 31) is I

=

const. Pick a solution linearly independent to this and multiply it by a constant so that its derivative is equal to the integrand in equation (2. 3 0). Then this is the required I(x).

Integrating the Laplace-transformed FP equation (2.14) and letting s ^-+0,

I

P (z Ix )dz

=

0 0

x n

lim q(x, s Ix )+H(x-x ) ,

s-+O o o (2.32)

where H(x) is the Heaviside unit function. This is an alternative method of finding I--but only useful if p(x, s jx ), and hence

q(x, s Ix ), has been found already.

2.3 THE LAPLACE-TRANSFORMED

Dalam dokumen Finally he would like to express his appreciation for the expert typing and patience of Mrs (Halaman 47-57)